Necessary and sufficient conditions for regularity of constraints in convex programming

This paper derives some necessary and sufficient conditions for (Lagrangian) regularity of the nondifferentiable convex programming problem. Furthermore, some weakest constraint qualifications are presented using the supporting functions and their derivatives, the outer normal cones, the single constraint function and its directional derivatives and epigraph and the projections of the outer normal cones     

Tangent cone and contingent cone to the intersection of two closed sets

For a nonempty closed set C in a real normed vector space X and an inequality solution set, we present several sufficient conditions for the tangent and contingent cones to their intersection to contain the intersections of the corresponding cones. We not only express the contingent cone to a solution set of inequalities and equalities by the directional (or Fréchet) derivatives of the active inequality constraint functions and the Fréchet derivatives of the equality constraint functions but also the tangent cone by the Clarke (or lower Dini, or upper Dini) derivatives of the active inequality constraint functions and the directional derivatives of the equality constraint functions. By using a simple property of the function d_C−d_C^c, we characterize these cones by the hypertangent and hypercontingent vectors to the set C. Furthermore, these results allow us to present new constraint qualifications for the Karush-Kuhn-Tucker conditions.     

Differential stability in non-Lipschitzian optimization

In this paper, upper and lower bounds are established for the Dini directional derivatives of the marginal function of an inequality-constrained mathematical program with right-hand-side perturbations. A nonsmooth analogue of the Cottle constraint qualification is assumed, but the objective and constraint functions are not assumed to be differentiable, convex, or locally Lipschitzian. Our upper bound sharpens previous results from the locally Lipschitzian case by means of a subgradient smaller than the Clarke generalized gradient. Examples demonstrate, however, that a corresponding strengthening of the lower bound is not possible. Corollaries of this work include general criteria for exactness of penalty functions as well as information on the relationship between calmness and other constraint qualifications in nonsmooth optimization.The author is grateful for the helpful comments of a referee.     

Directional derivatives of difference functions of nonsmooth max-functions

In this paper, we give an upper estimate for the Clarke-Rockafellar directional derivatives of a function of the form f - g, where f, g are max-functions defined by locally Lipschitz but not necessarily differentiable functions on a closed convex set in a Euclidean space. As an application, we give a sufficient condition for f - g to have an error bound.     

On an Extended Lagrange Claim

Lagrange once made a claim having the consequence that a smooth function f has a local minimum at a point if all the directional derivatives of f at that point are nonnegative. That the Lagrange claim is wrong was shown by a counterexample given by Peano. In this note, we show that an extended claim of Lagrange is right. We show that, if all the lower directional derivatives of a locally Lipschitz function f at a point are positive, then f has a strict minimum at that point.     

Asplund空间中方向上导数和序列法紧性质分析法则

研究了广义微分结构中的集合方向Mordukhovich法锥、集值映射的方向上导数,以及集合和集值映射的方向序列法紧性的分析法则. 基于集合方向Mordukhovich法锥的交集法则,在方向内半紧性假设下,建立了集合的方向Mordukhovich法锥、集值映射的方向上导数的分析法则.此外,借助Asplund乘积空间中集合的方向序列法紧性的交集法则, 在方向内半紧性和相应的规范条件下,建立了集合和集值映射的(部分)方向序列法紧性的加法、逆像、复合等法则.     

On Basic Constraint Qualifications for Infinite System of Convex Inequalities in Banach Spaces

The BCQ and the Abadie CQ for infinite systems of convex inequalities in Banach spaces are characterized in terms of the upper semi-continuity of the convex cones generated by the subdifferentials of active convex functions. Some relationships with other constraint qualifications such as the CPLV and the Slate condition are also studied. Applications in best approximation theory are provided.     

Farkas' theorem of nonconvex type and its application to a min-max problem

This note is concerned with the generalization of Farkas' theorem and its application to derive optimality conditions for a mix-max problem. Farkas' theorem is generalized to a system of inequalities described by sup-min type positively homogeneous functions. This generalization allows us to deal with optimization problems consisting of objective and constraint functions whose directional derivatives are not necessarily convex with respect to the directions. As an example of such problems, we formulate a min-max problem and derive its optimality conditions.The author would like to express his sincere thanks to Professors S. Suzuki and T. Asano of Sophia University and Professor K. Shimizu of Keio University for encouragement and suggestions.     

Second-order necessary efficiency conditions for nonsmooth vector equilibrium problems

This paper presents primal and dual second-order Fritz John necessary conditions for weak efficiency of nonsmooth vector equilibrium problems involving inequality, equality and set constraints in terms of the Páles–Zeidan second-order directional derivatives. Dual second-order Karush–Kuhn–Tucker necessary conditions for weak efficiency are established under suitable second-order constraint qualifications.     

On Constraint Qualifications in Nonsmooth Optimization

We introduce extensions of the Mangasarian-Fromovitz and Abadie constraint qualifications to nonsmooth optimization problems with feasibility given by means of lower-level sets. We do not assume directional differentiability, but only upper semicontinuity of the defining functions. By deriving and reviewing primal first-order optimality conditions for nonsmooth problems, we motivate the formulations of the constraint qualifications. Then, we study their interrelation, and we show how they are related to the Slater condition for nonsmooth convex problems, to nonsmooth reverse-convex problems, to the stability of parametric feasible set mappings, and to alternative theorems for the derivation of dual first-order optimality conditions.In the literature on general semi-infinite programming problems, a number of formally different extensions of the Mangasarian-Fromovitz constraint qualification have been introduced recently under different structural assumptions. We show that all these extensions are unified by the constraint qualification presented here.     

Directional derivatives of optimal value functions in mathematical programming

This note discusses the existence of the directional derivatives of the optimal value functions in a class of nonlinear programming problems and gives the expressions of the directional derivatives. In the study, it is not assumed that the optimal set at the point discussed is not empty. Many well-known results of this area can be derived as special cases of the main theorems of this note.This research was supported by the National Science Foundation of China. The authors would like to thank Professor A. V. Fiacco and the referees for their helpful suggestions.     

Necessary conditions for efficiency in terms of the Michel–Penot subdifferentials

In this article, we study a multiobjective optimization problem involving inequality and equality cone constraints and a set constraint in which the functions are either locally Lipschitz or Fréchet differentiable (not necessarily C ¹-functions). Under various constraint qualifications, Kuhn–Tucker necessary conditions for efficiency in terms of the Michel–Penot subdifferentials are established.     

Constraint qualifications in linear vector semi-infinite optimization

Linear vector semi-infinite optimization deals with the simultaneous minimization of finitely many linear scalar functions subject to infinitely many linear constraints. This paper provides characterizations of the weakly efficient, efficient, properly efficient and strongly efficient points in terms of cones involving the data and Karush–Kuhn–Tucker conditions. The latter characterizations rely on different local and global constraint qualifications. The global constraint qualifications are illustrated on a collection of selected applications.     

Abstract cone approximations and generalized differentiability in nonsmooth optimization

In the present paper a connection between cone approximations of sets and generalized differentiability notions will be given. Using both conceptions we present an approach to derive necessary optimality conditions for optimization problems with inequality constraints. Moreover, several constraint qualifications are proposed to get Kuhn-Tucker-type-conditions.     

A note on upper Lipschitz stability, error bounds, and critical multipliers for Lipschitz-continuous KKT systems

We prove a new local upper Lipschitz stability result and the associated local error bound for solutions of parametric Karush–Kuhn–Tucker systems corresponding to variational problems with Lipschitzian base mappings and constraints possessing Lipschitzian derivatives, and without any constraint qualifications. This property is equivalent to the appropriately extended to this nonsmooth setting notion of noncriticality of the Lagrange multiplier associated to the primal solution, which is weaker than second-order sufficiency. All this extends several results previously known only for optimization problems with twice differentiable data, or assuming some constraint qualifications. In addition, our results are obtained in the more general variational setting.     

多参数二次特征值问题重特征值的灵敏度分析

本文研究解析依赖于多参数的二次特征值问题重特征值的灵敏度分析,得到了重特征值的方向导数,证明了相应的特征向量矩阵和特征值平均值的解析性,给出了其一阶偏导数的表达式．然后以这些结论为基础,定义了二次特征值问题重特征值及其不变子空间的灵敏度,并给出了确定二次特征值问题所含矩阵中敏感元素的方法．     

Constraint qualifications for constrained Lipschitz optimization problems and applications to a MPCC

Three constraint qualifications (the weak generalized Robinson constraint qualification, the bounded constraint qualification, and the generalized Abadie constraint qualification), which are weaker than the generalized Robinson constraint qualification (GRCQ) given by Yen (1997) [1], are introduced for constrained Lipschitz optimization problems. Relationships between those constraint qualifications and the calmness of the solution mapping are investigated. It is demonstrated that the weak generalized Robinson constraint qualification and the bounded constraint qualification are easily verifiable sufficient conditions for the calmness of the solution mapping, whereas the proposed generalized Abadie constraint qualification, described in terms of graphical derivatives in variational analysis, is weaker than the calmness of the solution mapping. Finally, those constraint qualifications are written for a mathematical program with complementarity constraints (MPCC), and new constraint qualifications ensuring the C-stationary point condition of a MPCC are obtained.     

Existence and Boundedness of the Kuhn-Tucker Multipliers in Nonsmooth Multiobjective Optimization

Using the idea of upper convexificators, we propose constraint qualifications and study existence and boundedness of the Kuhn-Tucker multipliers for a nonsmooth multiobjective optimization problem with inequality constraints and an arbitrary set constraint. We show that, at locally weak efficient solutions where the objective and constraint functions are locally Lipschitz, the constraint qualifications are necessary and sufficient conditions for the Kuhn-Tucker multiplier sets to be nonempty and bounded under certain semiregularity assumptions on the upper convexificators of the functions.     

广义向量变分不等式和广义对偶模型

研究实Banach空间中带有不等式约束的非光滑向量优化问题(VP).首先,借助下方向导数引进了广义Minty型向量变分不等式,并通过变分不等式来探讨问题(VP)的最优性条件.接着,利用函数的上次微分构造了不可微向量优化问题(VP)的广义对偶模型,并且在适当的弱凸性条件下建立了弱对偶定理.